ERIC ED616201: Noticing Structural Thinking through the CRIG Framework of Mathematical Structure

2021·0.21 MB·English

by  ERIC

Noticing structural thinking through the CRIG
framework of mathematical structure
Mark Gronow
Central Queensland University
<[e-mail]>
Structural thinking skills should be developed as a prerequisite for a young person’s future
mathematical understanding and a teachers’ understanding of mathematical structure is
necessary to develop students’ structural thinking skills. In this study, three secondary
mathematics pre-service teachers (PSTs) learned to notice structural thinking through the
CRIG framework of mathematical structure. The framework consists of Connections,
Recognising patterns, Identifying similarities and difference, and Generalising and
reasoning. I report here on how the CRIG framework helped the PSTs’ notice structural
thinking.
To develop an ability to notice structural thinking, teachers must first of all be aware of
mathematical structure. Mason et al. (2009) defined mathematical structure as “the
identification of general properties which are instantiated in a particular situation as
relationships between elements or subsets or elements of a set” (p. 10). Stephens (2008)
described structural thinking as an awareness of how mathematical properties develop into
generalisations. Furthermore, Mason et al. (2009) promoted structural thinking as
understanding the concepts and knowing procedures to use and when solving mathematical
problems.
Varied theories exist about structure; as mathematical structure or structural thinking.
Wertheimer (1945) proposed that mathematical structure is knowing how a formula is
connected to a mathematical concept. Hiebert and Lefevre (1986) combined conceptual and
procedural knowledge as ‘proceptual’ thinking across mathematical processes. Stephens
(2008) defined ‘structure’ as synonymous with relational thinking (Skemp, 1976). Schwarz
et al. (2009) proposed that structural thinking is knowing the relationships and connections
between mathematical concepts.
The concept of structural thinking in mathematics is not clearly understood by many
teachers of mathematics (Richland et al., 2012). Mason et al. (2009) stated that teachers’
awareness of structural relationships transforms students’ thinking and disposition to
engage, they believe that teachers need to focus on structure so students can think
structurally. Research in teachers’ awareness of mathematical structure or structural thinking
is limited. Gronow et al. (2020) explored secondary mathematics teachers’ understanding
and use of mathematical structure. Their study investigated how teachers used mathematical
structure and encouraged structural thinking through components of mathematical structure:
Connections, Recognising patterns, Identifying similarities and differences, and
Generalising and reasoning. The four components, known as CRIG pedagogical framework
of mathematical structure developed during Gronow et al.’s (2020) study found teachers’
identified with structure but were not aware they were using it in their teaching. The CRIG
framework, in this study, is introduced to PSTs as an effective mechanism for learning to
notice structural thinking. The four components of the CRIG framework are detailed next.
Connections. Vale et al. (2011) recognised connections between mathematical
representations as fundamental to structural thinking. Making connections between contexts
or concepts allows learners to develop mathematical understanding. Mathematics teachers
2021. In Y. H. Leong, B. Kaur, B. H. Choy, J. B. W. Yeo, & S. L. Chin (Eds.), Excellence in Mathematics
Education: Foundations and Pathways (Proceedings of the 43rd annual conference of the Mathematics
Education Research Group of Australasia), pp. 211-218. Singapore: MERGA.
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make connections between prior, present and future learning, and in real-world contexts of
mathematical representations.
Recognising patterns. Patterns are essential in children’s mathematical development
which begin with their observations of the natural world. Children recognise, observe and
generate patterns before reaching school and learn patterning in formalised learning
situations that develop structural thinking processes that lead to a deeper understanding of
abstract mathematical concepts. Mulligan and Mitchelmore (2009) found that children’s
structural thinking, identified in patterning awareness, is essential for mathematical concept
formation in future learning.
Identifying similarities and differences. Learners develop structural thinking through
noticing the differences in mathematical representations. Mason (1996) believed structural
thinking is noticing similarities and differences in mathematical relationships. Mulligan and
Mitchelmore (2009) discovered that children who found similarities and differences in
patterns were involved in structural thinking.
Generalising and reasoning. Mason (2008) described this as an activity that develops a
more in-depth experience of mathematics. Mathematical thinking that eventuates into a
generalised fact is structural thinking, it connects mathematical relationships from concrete
representations to abstract ideas. Mason et al. (2009) wrote that appreciation of structure
involves the experience of generality. Stephens (2008) applied structural thinking to
designing arithmetic questions. He asserted that children who could articulate a generalised
principle underlying a whole problem were thinking structurally.
The framework of noticing also supports the process PSTs learning to notice structural
thinking. Scheiner (2016) identified how noticing is not restricted to a single process. Mason
(2002) asserted that “every act depends on noticing” (p. 7), he used the term “awareness” to
characterise the ability to notice, referring to noticing as an awareness of what one is
attending to. In this study, noticing structural thinking implies an awareness of understanding
and use of mathematical structure.
By adopting Mason’s (2002) approach to noticing, the development and use of
mathematical structure has emerged as a form of directing PSTs’ attention to their
mathematical thinking. Mason studied what he noticed when doing mathematics and called
what he noticed the structures of attention of how one thinks mathematically. The aim of
this study is for the PSTs to notice structural thinking through learning the components of
the CRIG framework of mathematical structure. The PSTs use of the CRIG framework
provides an opportunity to detect their awareness of structure, thus answering the research
question: How does the CRIG framework help PSTs to notice structural thinking?
Method
Context and participants
PSTs in their final year Bachelor of Education/Bachelor of Arts (secondary mathematics)
degree at a Sydney university were invited to participate in this study. Three PSTs, referred
to as Ms K, Ms M, and Mr T, volunteered to participate in the study during their professional
experience placement. Each PST taught mathematics at a secondary school in metropolitan
Sydney. Ms K taught an accelerated Year 9 class, Ms M taught a top streamed Year 8 class,
and Mr T taught a mixed ability Year 7 class. The PSTs were familiar with the concept of
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mathematical structure through the content of courses studied in their undergraduate degree;
however, they had no prior knowledge of the CRIG framework.
Study design, instruments, and data collection
The study design comprised of three cycles of: professional learning workshops (PLWs),
which were audio recorded. Video recordings of PSTs’ mathematics lessons and a noticing
reflection audio recording of PSTs reviewing a recorded segment of their mathematics
lessons.
Analysis
The audio recordings of the PLWs and noticing reflections were all transcribed to a word
document and uploaded to NVivo (QSR International, 2017). The videos of the mathematics
lessons were also uploaded to NVivo. NVivo was used to code the data from the PLWs,
mathematics lessons and noticing reflections for PSTs’ utterances and comments that
identified a CRIG component. The data were analysed for evidence of the PSTs’ noticing of
structural thinking through the PSTs attending to the CRIG framework. The videos acted as
the main source of evidence for identifying the PSTs noticing structural thinking through
their use of the CRIG framework when teaching. The PLWs and the noticing reflections
provide further evidence of the PSTs attention to the CRIG framework.
Results
This section presents a summary of the data collected for each PST from the three cycles
of PLWs, mathematics lessons and noticing reflections. An outline of the results from the
PLWs are given, followed by exemplars of each PSTs’ utterances from the mathematics
lessons and comments made during the noticing reflections in Tables 1, 2 and 3, coded to a
CRIG component.
During the PLWs, the PSTs were taught to notice structural thinking through the CRIG
components. The first PLW began with a presentation on the CRIG framework, followed by
a viewing of a video titled Related Problems: Reasoning About Addition (Teaching Channel,
2017), where a teacher used the CRIG components to teach addition to a Year One primary
class. Ms K Recognised patterns in the teacher’s instructions to students. Ms M also
Recognised patterns as a teaching strategy to engage the students. Mr T noticed that the
students used Similarities and differences to make generalisations.
In PLW 2, the PSTs viewed a video recording of a child attempting several different
arithmetic problems, they were asked to examine the child’s mathematical thinking when
solving the problems. Ms K noted the child relied on calculations and did not Identify
Similarities and Differences between the numbers. Ms M noticed the child was using
Generalising and Reasoning in her structural thinking when she recognised that the problem
could be solved another way. Mr T stated the child “Got it after the CRIG prompt, meaning
she has structural understanding.”
In PLW 3, the PSTs considered how the CRIG framework could be applied to teaching
the expansion of binomial products. Ms K made Connections to the distributive law and
expanding the expression using the FOIL method. Ms M was Identifying Similarities and
Differences when changing numbers, pronumerals, signs and coefficients in the binomial
expression. Mr T stated that Generalising and reasoning was identified as a way to
summarise the process of expansion and apply it in other mathematical contexts.
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Table 1
Exemplars of Ms K using the CRIG Framework to Notice Structural Thinking
Cycle Mathematics lesson Noticing reflection
1 Topic: Simultaneous equations Connections between the equation
Connections to the relationship between the and the graph. “I think to show how
graphs’ intersection points and solving the the y2 and the x2 is giving us part of
equations simultaneously. the circle, that relationship.”
Recognising Patterns of the power of x to Identifying similarities and
differences between graphs and
determine the curve’s shape.
equations: “So, they could see that
Identifying similarities and differences
all of them had a square except the
“What is different about the line’s shape?”
last one.”
2 Topic: Angle sum of polygons Recognising Patterns to develop
Connections to prior learning “How did we the formula: “They understood it
prove the angle sum of the quadrilateral?” better with the pattern.”
Angle sum of a polygon formula: Identifying similarities and
Recognising patterns: “Can you find the differences different patterns helped
pattern of what is going on between the students’ thinking. “I had the
number sides and triangles?” triangles meeting at a point. I
adjusted it as I saw the pattern they
Generalising and reasoning: “Calculate the
were working out.”
interior angle sum of any polygon.”
3 Topic: Quadratics Connections: “I was connecting it
Connections “Quadratics and parabolas go to when we did the non-linear
hand-in-hand. The visual representation of a simultaneous equations.”
quadratic is a parabola.” Recognising Patterns, “Rather than
Identifying Similarities and Differences of drawing random graphs, I’d link
the x2 expression in an equation “This is not them to recognise any patterns from
of degree two; it is a power of negative two. factorised quadratics.”
So, this is not a quadratic.” Generalising and Reasoning
“Generalising the solutions of when
Generalising and Reasoning relationships
between the equation and the graph. crossing the x-axis.”
Table 2
Exemplars of Ms M using the CRIG Framework to Notice Structural Thinking
Cycle Mathematics lesson Noticing reflection
1 Topic: Circumference of a circle Generalising and Reasoning through
Connections to a real life example of a students’ discussion when dividing
pizza as a sector of a circle. the circumference by the diameter.
“I’m looking at what they just did.
Recognising patterns in the ratio of a
I’m asking them to contribute what
circle’s circumference and diameter.
they found and see what they
Similarities and Differences comparing
conclude from what they've done.”
the circle’s radius and diameter.
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2 Topic: Area of composite shape Recognising patterns “asking them
Identifying Similarities and differences to how to figure out the area. That
explain the formula of the area of circles. could have been kind of recognising
“Area equals 𝜋𝑟2 which is the same as patterns.”
saying 𝜋×𝑟×𝑟.” Identifying Similarities and
differences “How to write something
Generalising and reasoning “How come
in exact form and not exact form
we have 𝜋 for every circle? Because the
circumference divided by the diameter Generalising and reasoning “Asking
was always equal to 𝜋.” them questions they can conclude.”
3 Topic: Volume of a cylinder Connections “How they could use
Connections of a real-world problem: previous things they've learnt.”
“This is a picture of the sinkhole. What Recognising patterns “By helping
shape does it look like?”, them recognise patterns to work
Generalising and reasoning “What do we mathematically.”
need to know to solve this problem? What Generalising and reasoning
are we trying to find in the end?” “Recognising the meaning and
interpreting the information.”
Table 3
Exemplars of Mr T using the CRIG Framework to Notice Structural Thinking
Cycle Mathematics lesson Noticing reflection
1 Topic: Ordering fractions Connections “I should have
Connections to a real example “What is reworded the question because this
one-third of my chocolate bar.” was what we did last lesson.”
Identifying Similarities and Differences in Recognising Patterns “What do you
ordering fractions “When you look at this, notice I’m doing with these
which one’s bigger? Or, which one’s numbers?”
smaller?” Identifying similarities and
Generalising and Reasoning defining a differences “Show the diagram of
rule “The size of the parts needs to be the shaded fractions not symbolically.”
same.”
2 Topic: Adding and subtracting fractions Recognising patterns: “I tried to set
Identifying Similarities and Differences up some patterns and then asked
“What do you notice about the them to recognise the patterns.”
numerators?” Generalising and Reasoning “I’ve
Generalising and Reasoning, using a tried to incorporate generalisation in
whole number method to add fractions. terms of asking them, ‘What do you
think would be the next pattern?’”
“So, if 1+1 = 2, then, if I use the same
1 1
thing, for a + , is 1+1 = 2, and 2+
2 2
1
2 = 4, so it’s over . Right?”
4
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3 Topic: Stem and leaf plot graphs Recognising patterns: “So I should
Similarities and Differences between have put one number on so the
graphs and stem-and-leaf plots. “Now students to see a pattern.”
what were the things we compared. Identifying Similarities and
What’s similar?” Differences “I should have asked
Generalising and reasoning to analyse about the placement of these three
stem-and-leaf plot data. “Take a look at numbers: “How are they different?”
your graph and talk to the other person
and tell them what the graph tells you?”
Discussion
During this study, the PSTs’ noticing of structural thinking developed through their
learning of the CRIG pedagogical framework of mathematical structure. Noticing of
structural thinking was evident in their references to the CRIG framework drawn from the
statements made during the PLWs, utterances in their mathematics lessons, and noticing
reflection comments. Exemplars given demonstrate the PSTs’ noticing structural thinking
through the CRIG components.
The PSTs use of the CRIG components were identified in varied pedagogical strategies.
Ms K encouraged students to use a pattern to find the rule for the angle sum of a polygon,
Ms M used real world examples for each of her lessons to connect students understanding to
the mathematical concept and Mr T used the CRIG components in his questions.
The PSTs’ teaching accommodated the CRIG framework and supported their
understanding of the mathematical content. Ms K considered other patterning approaches to
finding a rule for the angle sum of a polygon and Ms M noticed similarities and differences
in binomial expansions. The PSTs’ pedagogy focused on a structural thinking learning
environment, Ms K promoted students’ thinking by challenging them to connect the equation
to a graph, Ms M connected mathematical concepts to real-world examples and Mr T asked
questions so students would notice patterns, and similarities and difference. In their noticing
reflections, the PSTs stated how the CRIG framework supported their teaching. Ms K, was
thinking of her future teaching: “If I were to do this again, I’d teach the patterning way, and
I would incorporate the CRIG more.” Ms M stated CRIG helped her understand student
thinking “They're trying to understand the difference between volume and capacity.” Mr T
reflected on how CRIG improved his explanations. “I should have made it more explicit, by
connecting to their prior experience.” The CRIG framework in these cases supported the
PSTs’ noticing of structural thinking.
Prescott and Cavanagh (2007) found that secondary mathematics PSTs tended toward a
traditional teaching pedagogy. Awareness of the CRIG framework encouraged the PSTs in
this study to move beyond traditional teaching pedagogy. The PSTs were more inclusive of
student learning, as noted when asking CRIG component focused questions. Mr T’s
questions promoted students’ structural thinking. He challenged students’ thinking about
why using a whole number method when adding fractions was incorrect. “So, if 1+1 = 2,
1 1 1
then, if I use the same thing, for a + , is 1+1 = 2, and 2+2 = 4, so it’s over . Right?”
2 2 4
The PSTs diverse pedagogical strategies also saw them use the CRIG components when
instructing or communicating with students. In her second mathematics lesson, Ms K used
Recognising patterns to help students develop the angle sum of a polygon formula. As the
students had discovered a different pattern, one that was not considered by Ms K, she acted
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in-the-moment and noticed the students’ new approach, she encouraged her students to
continue with their strategy and asked one student to explain it to the class. Ms M promoted
student involvement in her lessons by arranging students in groups to complete activities,
many of which had a real-world experience, such as, here final lesson of finding the volume
of a cylinder as a sink hole.
The professional learning program to understand and use the CRIG framework helped
the PSTs’ to notice structural thinking. Ivars et al. (2018) identified the need for a specific
framework for PSTs to have effective noticing. The CRIG framework provided this focus.
The ability of the PSTs to understand the CRIG framework and to use it demonstrated its
simplicity as a practical and useful tool for teachers of mathematics. The PSTs’ content
knowledge was established from their extensive mathematical background in their university
studies. The CRIG framework, however, deepened the PSTs structural understandings of
mathematical relationship, for example Ms K’s students finding an alternative approach to
finding the angle sum of a polygon.
The PSTs’ lack of professional experience before this study could have influenced their
fundamental understanding of the CRIG framework and their ability to notice structural
thinking. However, having more teaching experience in the future will provide continual
opportunities notice structural thinking through the CRIG framework when doing
mathematics and when teaching. The PSTs’ teaching experience was restricted to their
university professional experience program. Researchers have identified how PSTs’ limited
experiences influence what they attend to when teaching. Star and Strickland (2008) found
that secondary mathematics PSTs were not good at noticing mathematical content. Mason
(2002) also asserted that PSTs lack experience in recognising and using classroom
interactions effectively to promote mathematical understanding. Contrary to the results of
these studies, the PSTs in this study produced mathematics lessons that engaged students
with activities, instructions and questions that focused on developing students’ structural
thinking through using the components of the CRIG framework. The PSTs effectively
demonstrated an ability to learn and apply the CRIG framework as a new pedagogical skill
to mathematical content that they had not taught before. The introduction of structural
thinking through the CRIG framework could be regarded as an extra burden for the PSTs to
consider when teaching. Nevertheless, the evidence indicates that the PSTs were comfortable
with identifying and including the components of the CRIG framework in their lessons and
were able to notice structural thinking.
The PSTs were able to articulate the benefits of the CRIG framework to notice structural
thinking they indicated that the CRIG framework had shaped their noticing structural
thinking and had changed their teaching. Ms K stated that thinking structurally helped her
make sense and explain mathematical concepts. In the final PLW, Ms K stated, “You
structure your practice to facilitate deeper thought as to what and how things made sense.”
Conclusions and Further Research
The CRIG framework proved to be useful for helping PSTs to notice structural thinking.
The CRIG framework provided the PSTs with a foundation for teaching mathematics that
helped them focus on developing their understanding of mathematical structure. Moreover,
this provided PSTs opportunities to notice structural thinking.
Mason (2002) introduced the concept of noticing into the lexicon of mathematics
education, and with his colleagues (Mason et al. 2009) the notion of teachers’ noticing of
structural thinking has emerged as a significant contribution to mathematics teaching. PSTs
noticing of structural thinking as the focus of this study has demonstrated, as evident from
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the results, that there is potential to advance the discourse of mathematics teaching in this
area.
The introduction of mathematical structure in the teaching and learning of mathematics
and the noticing of structural thinking has implications for future research in mathematics
teaching. Future research could consider how developing noticing structural thinking
through the CRIG framework may benefit practicing teachers of mathematics (e.g., primary,
secondary, pre-service, novice, experienced, and out-of-field teachers).
References
Gronow, M., Mulligan, J., & Cavanagh, M. (2020). Teachers’ understanding and use of mathematical structure.
Mathematics Education Research Journal, 1-26. https://doi.org/10.1007/s13394-020-00342-x
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory
analysis. In J. Hierbert (Ed.), Conceptual and procedural knowledge: The case of mathematics. Hillsdale,
NJ: Erlbaum
Ivars, P., Fernández-Verdú, C., Llinares, S., & Choy, B.H. (2018). Enhancing noticing: Using a hypothetical
learning trajectory to improve pre-service primary teachers’ professional discourse. Eurasia Journal of
Mathematics, Science and Technology Education, 14(11). https://doi.org/10.29333/ejmste/93421
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.)
Approaches to Algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht: Kluwer Academic
Publishers.
Mason, J. (2002). Researching your own practice: The discipline of noticing. London: RoutledgeFalmer.
Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. In J. Kaput, D. Carraher, &
M. Blanton (Eds.), Algebra in the early grades (pp. 57–94). Mahwah, NJ: Lawrence Erlbaum.
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics
Education Research Journal, 21(2), 10–32.
Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical
development. Mathematics Education Research Journal, 21(2), 33–49.
Prescott, A., & Cavanagh, M. (2007). Pre-service secondary mathematics teachers: A community of practice.
In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.) Proceedings of the Conference of the International
Group for the Psychology of Mathematics Education. (pp. 276–284). Seoul, Korea: Society of Educational
Studies in Mathematics.
QSR International. (2017). NVivo 12 [Computer software]. http://www.qsrinternational.com.
Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics.
Educational Psychologist, 47, 189–203.
Scheiner, T. (2016). Teacher noticing: Enlightening or blinding? ZDM – Mathematics Education, 48, 227–238.
Schwarz, B. B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in
context. In B. B. Schwarz, T. Dreyfus & R. Hershkowitz (Eds.), Transformation of knowledge through
classroom interaction (pp. 11–41). New York, NY: Routledge
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77,
20–26.
Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve preservice mathematics
teachers’ ability to notice. Journal of Mathematics Teacher Education, 11(2), 107–125.
Stephens, M. (2008, September). Designing questions to probe relational or structural thinking in arithmetic.
Paper presented at the 5th Annual Conference of the International Society for Design and Development in
Education, Cairns.
Teaching Channel. (2017). Related problems: Reasoning about addition. Retrieved from
https://www.teachingchannel.org/video/reasoning-about-addition-nsf
Vale, C., McAndrew, A., & Krishnan, S. (2011). Connecting with the horizon: Developing teachers’
appreciation of mathematical structure. Journal of Mathematics Teacher Education, 14(3), 193–212.
Wertheimer, M. (1945) Productive thinking. London: Tavistock.

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